## Abstract

The algebra generated by the four matrices $\beta _{\mu}$ occurring in the relativistic wave equation of a particle of maximum spin n on the basis of the commutation rules for these matrices obtained previously by one of the authors has been investigated. Auxiliary quantities $\eta _{\mu}$ satisfying the equations (5) are introduced. These $\eta _{\mu}$ are given as polynomials in $\beta _{\mu}$. With the help of these, further auxiliary quantities $\xi _{\mu}$ = $\eta _{\mu}\beta _{\mu}$ are defined. It is shown that for half odd integral spin, the $\xi $'s and $\eta $'s form two mutually commuting sets of symbols of which the $\eta $'s satisfy the same commutation rules as the Dirac matrices. This proves that the algebra in the case of half odd integral spin is the direct product of the Dirac algebra and an associated $\xi $-algebra. For the special case of maximum spin $\frac{3}{2}$ the $\xi $-algebra has been studied in detail, and it is shown that this algebra has just three representations of orders 1, 4, 5 such that 1$^{2}$ + 4$^{2}$ + 5$^{2}$ = 42 = rank of the algebra. Explicit representations are given in the non-trivial cases of orders 4 and 5. The 4-row representation of the $\xi $'s gives a representation of the $\beta $'s of order 16 which is likely to be of importance in connexion with Bhabha's new theory of the proton.